It depends on where you're located. Some areas don't have the good hardwoods that make the best fires ( long lasting and cleaner burning. ) Oak, hickory, locust, maple, birch, cherry, hop-hornbeam are the most common good hardwoods. Ash is also a good one, and many are dying because of the emerald ash borer ( EAB ). The dead ash will provide lots of good firewood for many folks.
Like Tooln said above, make sure it's dry. The rule of thumb is to season it for a year before burning. I cut my wood at least a year ahead of use. Unseasoned wood smolders and burns a smoky fire, causing creosote to build up in the chimney - the fuel of chimney fires. If you see someone's chimney constantly smoking, they either have wood that's too green or they don't know how to regulate their fire/woodstove. Hardwoods have better " coaling properties " - meaning they make good hot coals that last. I use all of the ones I mentioned above except locust. No locust near where I cut my wood. Whatever you decide to burn - dry it first !! ( I grew up with coal furnace and woodstove - have woodstove insert in my home fireplace to this day. )